Radicals and rational exponents

So far, we have encountered integer-type exponents. In this chapter, we will explore fractional exponents and their representation as radicals. A fractional exponent is an exponent on a variable or number that is not a whole number, like $x^{\frac{1}{2}}$.

Denominator of fractional exponents

Square root

A fractional exponent can be seen as the opposite of an integer exponent. For instance:

$$4^2=16$$

And:

$$4^{\frac{1}{2}}=2$$

You can see here that when the $2$ is in the numerator of the exponent, $\frac{2}{1}$, the number is squared, making $4\ast 4=16$. Conversely, when the $2$ is in the denominator of the exponent, $\frac{1}{2}$, the opposite happens where we end up with the number that would make $4$ when squared. That number is $2$ in this case: $2\ast 2=4$.

This is what we refer to as the square root. Maybe you already knew that the square root is the opposite of a squared number, but what we will discover is that a root, also known as a radical, is the opposite of any exponent.

Cube root

We know that a cubed number is a number to the power of $3$. So, the opposite of this exponent would be a number to the power of $\frac{1}{3}$. Much like the square root, the fractional exponent $\frac{1}{3}$ can be represented by a radical and called the cube root. But, the cube root has a $3$ in front of the radical to specify the denominator of the fractional exponent. Take a look at the difference between the square root and cube root.

$$x^{\frac{1}{2}}=\sqrt{x}$$

$$x^{\frac{1}{3}}=\sqrt[3]{x}$$

You see that there is a $3$ in front of the radical for the cube root, though there is not a $2$ in front of the radical for the square root. It is a little inconsistent, but this is the convention for radicals. If there is no number in front of the radical, it is a square root. If there is a number in front of the radical, then that number is the denominator of the fractional exponent. Likewise, if there is a number in the denominator of a fractional exponent other than $1$ or $2$, that number goes on the outside of the radical, attached to the radical itself. This number in front of the radical is called the index.

Numerator of fractional exponents

Fractional exponents will not always have a $1$ in the numerator. When there is an integer in the numerator of a fractional exponent, that number goes inside the radical, attached to whatever is inside. Let’s see the example $x^{\frac{2}{3}}$:

$$x^{\frac{2}{3}}=\sqrt[3]{x^2}$$

Here, the $2$ goes inside the radical attached to the $x$.

Now you have everything you need to convert a fractional exponent into a radical. You will occasionally be asked to do this because fractional exponents are confusing and undesirable in a math expression. You can consider radicals as the simplified form of fractional exponents.

Conversion practice

You need to be comfortable converting from a fractional exponent to a radical and from a radical to a fractional exponent. These are core concepts frequently tested. So, we will go through several examples below.

What is $\sqrt[4]{x^2}$ as an exponent?

The numerator of the fractional exponent is the number inside the radical, $2$. The denominator of the fractional exponent is the number attached to the outside of the radical, $4$.

So, the exponent form of this expression is $x^{\frac{2}{4}}$ which can be further simplified to $x^{\frac{1}{2}}$

What is the exponent form of $\sqrt[3]{x^3}$?

What is $x^{\frac{5}{2}}$ expressed as a radical?

What is $x^{-\frac{2}{3}}$ expressed as a radical?